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Complex Analysis and Operator Theory

, Volume 13, Issue 8, pp 3853–3870 | Cite as

A Class of Hausdorff–Berezin Operators on the Unit Disc

  • Alexey KarapetyantsEmail author
  • Stefan Samko
  • Kehe Zhu
Article
  • 106 Downloads

Abstract

We introduce and study a class of Hausdorff–Berezin operators on the unit disc based on Haar measure (that is, the Möbius invariant area measure). We discuss certain algebraic properties of these operators and obtain boundedness conditions for them. We also reformulate the obtained results in terms of ordinary area measure.

Keywords

Hausdorff operators Berezin transform Bergman kernel Invariant kernel Haar measure Möbius group 

Mathematics Subject Classification

47G10 47B38 46E30 

Notes

Acknowledgements

Alexey Karapetyants acknowledges the support of the Fulbright Research Scholarship program and the warm hospitality of the Mathematics Department at the State University of New York at Albany during the time when this research was completed. Funding was provided by J. William Fulbright Research Scholarship Program (Grant No. PS00267032). Alexey Karapetyants and Stefan Samko are partially supported by the Russian Foundation for Fundamental Research (Grant Number 18-01-00094). Kehe Zhu is supported by the National Natural Science Foundation of China (Grant Number 11720101003) and by STU Scientific Research Foundation for Talents (Grant No. NTF17009).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSUNYAlbanyUSA
  2. 2.Department of Mathematics and Regional Mathematical CenterSouthern Federal UniversityRostov-on-DonRussia
  3. 3.University of AlgarveFaroPortugal
  4. 4.Department of MathematicsShantou UniversityShantouChina

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